Regular semi-algebraic system

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In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.

Introduction[edit]

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems.

Any semi-algebraic system can be decomposed into finitely many regular semi-algebraic systems such that a point (with real coordinates) is a solution of if and only if it is a solution of one of the systems .[1]

Formal definition[edit]

Let be a regular chain of for some ordering of the variables and a real closed field . Let and designate respectively the variables of that are free and algebraic with respect to . Let be finite such that each polynomial in is regular w.r.t.\ the saturated ideal of . Define . Let be a quantifier-free formula of involving only the variables of . We say that is a regular semi-algebraic system if the following three conditions hold.

  • defines a non-empty open semi-algebraic set of ,
  • the regular system specializes well at every point of ,
  • at each point of , the specialized system has at least one real zero.

The zero set of , denoted by , is defined as the set of points such that is true and , , for all and all . Observe that has dimension in the affine space .

See also[edit]

References[edit]

  1. ^ Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao. Triangular decomposition of semi-algebraic systems. Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187–194, 2010.